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Mathematics in Ancient and Medieval India c6.s4: Theorem of Square on the Diagonal

Mathematics in Ancient and Medieval India c6.s4

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Theorem of Square on the Diagonal

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The theorem is enunciated by Baudhāyana (c. 600 B. C. ) as follows :

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*dīrghacaturaśrasyākṣṇayārajjuḥ pārśvamānī tiryaṅmānī ca yatpṛthagbhūte kurutastadubhayaṃ karoti*।

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“The diagonal of a rectangle produces both (areas) which its length and breadth produce separately”. That is, in a rectangle ABCD, AC² = AB² + BC². This is a most general statement and was enunciated first by Baudhāyana. The proposition is stated in almost identical terms also by Āpastamba and Kātyāyana². Baudhāyana further says that the theorem is easily verified from the following relations :

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$3² + 4² = 5²$ $12² + 5² = 13²$ $15² + 8² = 17²$ $7² + 24² = 25²$ $12² + 35² = 37²$ $15² + 36² = 39²$

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² Datta, B. — *The Science of Śulba*, p. 104-132, Calcutta University, 1932.

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No proof of this theorem is given by Baudhāyana and other Śulba writers, since it is beyond their traditions. Zeuthen¹, Cantor², Vogt³, Cajori⁴, and Heath⁵ have expressed the view that the general statement was possibly the result of an induction from a small number of cases of right-angled triangles having sides in rational numbers known to them. But this is not the actual case. Our discussion on rational rectangles and construction of geometrical figures amply justify that the general character of the theorem was rightly understood by the Sulbakāras. This has been discussed by Burk⁶, who concluded that the theorem of square on the diagonal was known and proved in all its generality by the Indians long before the date of Pythāgoras (c-540 B. C ). He went further that so far from Indian Geometry being indebted to the Greek, the much travelled Pythāgoras probably obtained his theory from India.⁷ In this connection a number of conjectures by Bretschneider¹, Gow², Hankel³, Allman⁴, Heath⁵, Dutta⁶ and others as to the way the proof of the theorem could have been arrived at by the Indians are available.

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¹ Zeuthen, H. G. 'Theorem de Pythagore Origine de la geometrie scientifique', *Comptes Rendus du IIme Congress internationale de Philosophie*, Geneve, 1904; 'Sur l' Arithmetique Geometrique des Grecs et des Indiens', *Bibliotheca Mathematika*, Series 3, 5, p. 97-112, 1904.

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² Cantor, M. 'Uber die alteste indische Mathematique,' *Archiv der Mathematik und Physik*, 8, p. 63-92, 1905.

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³ Vogt, H. 'Haben die alten Inder den Pythagorischen Lehrsatz und das Irrationale Gekannt ?', *Bibliotheca Mathematica*, 7, pt. 3, p. 6-23, 1906.

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⁴ Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 86, 1919.

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⁵ Heath. T. L. *A History of Greek Mathematics*, 1, p. 147, Oxford, 1921.

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⁶ *Zeitschrift der deustschen morgenlandischen Gesellschaft (ZDMG)*, 55, p. 546.

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⁷ *ZDMG*, 55, p. 575; Jones, Sir William, *Works*, 8, p. 236; Colebrooke, T. *Miscellaneous Essays*, *1*, p. 436; Garbe R. *Philosophy of Ancient India*, p. 39 etc.

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¹ Bretschneider, C. A. *Die Geometrie und die Geometer von Eukleīdes?*, p. 82, Leipzing, 1870.

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³ Hankel, H. *Zur Geschichte der Mathematik in altertum in mitelalter*, p. 98, Leipzig, 1874.

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⁴ Allman, G. *Greek Geometry from Thales to Euclid*, Dublin, p. 37, 1889.

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⁵ Heath, T. L. *The Thirteen Books of Euclid's Elements*, Cambridge, *1*, p. 352, 1908.

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⁶ Datta, B. *The Science of Śulba*, p. 159, 1932.

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The theorem of square on the diagonal of a rectangle i.e. a² + b² = c², is usually known as Pythagorian theorem and is universally associated with the name of the Greek Pythagoras (c. 540 B C.). In fact the relation 3² + 4² = 5² and some such relations have been used by Pythagoras, and the evidence of any general statement regarding this is not yet available. Actual proof was first given by Euclid (c. 300 B. C.).⁷ Proclus (c 460 A. D.), the commentator of Euclid remarked : "for my part, while I admire those who first observed the truth of the theorem, I marvel more at the writer of the Elements, not only because he made it first by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth book.¹ Heath admitted with the remarks : “it is difficult for us to be more positive than Proclus was”.² Hence it is far from conclusive that Pythagoras stated the above general theorem. Variety of evidence is at present available that the practical use of the theorem was current in old Babylonian times (c. 1800–1600 B. C.). The evidence for this is found in certain Babylonian cuneiform tablets. No general theorem was found to have been mentioned. It has been conclusively proved by Neugebauer that Pythagoras has derived his “Number theorem of Universe” as well as the so called Pythagorian Theorem from the Babylonian cuneiform tablets³. The Chinese knew a similar relation which appeared in *Chou Pei* (4th century B. C.) and the most of its credit is connected from the time of its first commentator Chao Chun Chhing (3rd century A. D.).⁴ In India, the proof first appeared in the work of Bhāskara II⁵ (c. 1150 A. D.). Needham remarks “Bhāskara’s treatment derives from the Chou Pei”.⁶ This is not however true, for the proofs of Bhāskara II and that stated to be of *Chou Pei* can readily be deduced from a number of constructions described in the *Śulbasūtras*¹.

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² Gow, J. *A Short History of Greek Mathematics*, Cambridge, p. 155, 1884.

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⁷ *Elements*, Book 1, Proposition 47.

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¹ Proclus’ commentary on the Euclid, Book I, is one of the two main sources of information as to the History of Greek Geometry, the other being the collection of Pappus.

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² Heath, T. L. *A Manual of Greek Mathematics*, Dover Publications, p. 96, 1963.

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³ Neugebauer, O. *The Exact Sciences in Antiquity*, p. 28-42, 1952.

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⁴ Needham, J. *Science and Civilization in China*, 3, p. 95.

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⁵ *Bijagaṇita* of Bhāskara II. *Vide* Sudhākara Dvivedi’s edition, p. 70.

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⁶ Needham, J. *Ibid.*, 3, p. 19.

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¹ Datta, B. *The Science of Śulba*, p. 77, University of Calcutta, 1932.

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